Let $S$ be a set of $n$ elements. The number of ordered pairs in the largest and the smallest equivalence relations on $S$ are:
Refer --> https://math.stackexchange.com/questions/1081333/prove-that-the-empty-relation-is-transitive-symmetric-but-not-reflexive
Smallest Equivalence relation on set S = ∆ ( Diagonal relation )
Number of elements in Diagonal relation = ∣S∣ = n
So, cardinality of Smallest Equivalence relation on set S = n
Note: Empty relation on Non-empty set will never be an equivalence relation because it does not satisfy the reflexive property.Whereas Empty relation on empty set will always be an equivalence relation.
Largest Equivalence relation on set S = S ⨉ S = ∣ S ⨉ S ∣ = n^2
So, cardinality of Largest Equivalence relation on set S = n^2